'Mixing Lemonade' printed from http://nrich.maths.org/

Lots of great solutions were submitted to this
problem, using a wide variety of approaches. The problem prompted
you to use fractions, ratios, percentages, and graphs. In addition,
you could investigate and consider which methods worked most
effectively in different situations.

The first part of the task is to
determine the mixture with the stronger tasting lemonade
-

the first
glass has $200$ml of lemon juice, and $300ml$ waterthe second glass has $100$ml of lemon
juice, and $200$ml water

What is the best way to compare these two
mixtures? There are several different ways, as shown by the
different explanations submitted.

Mahir, from Saltus Grammar School
converted the values given so that there is the same amount of
water in each glass:

In the first glass, there is $300$ml water,
and in the second, there is $200$ml. Therefore, to equate the
amounts of water, he multiplied everything in the first glass by
two, and everything in the second glass by three. Each glass now
has $600$ml water, and so can now be compared. Note that it is "ok"
to multiplyeverythingin each
glass by a certain number as it is theproportionsorrelativeamounts
we are interested in, rather than the absolute amounts.

For the first glass, we now have:

$400$ml lemon juice
and $600$ml water

For the second glass, we now have:

$300$ml lemon juice
and $600$ml water

From this, we can now tell that the mixture in
the first glass must taste stronger: for the same amount of water,
there is more lemon juice.

Jonathan, from Wilson's School, used a similar
method. However, he instead made the amounts of lemon juice equal,
and then saw which glass had more water. The glass with less water
for the same amount of juice will be stronger, as the juice is less
diluted.

Another related method, used by many people
was to use ratios, fractions and/or percentages. Will K. from
Wilson's School gave a lovely explanation:

Glass 1 would be stronger as it has a simplified ratio of lemon
juice: water of $2:3$, and so is $\frac{2}{5}$ lemon juice, or
$40$%, whereas glass $2$ has a simplified ratio of $1:2$ for lemon
juice: water, and so is $\frac{1}{3}$ lemon juice, or $33.\dot{3}$
% lemon juice, and so is weaker than glass $1$.

The strategy is to work out the ratio (e.g. $40$ml lemon juice to
$120$ml water would be $40:120$ lemon juice: water), simplify it
(e.g. $40:120$ simplifies to $4:12$, which simplifies to $1:3$),
and then turn that into a fraction ($1:3$ would be $\frac{1}{4}$ as
$1 +3 = 4$ and so the $1$ part that is lemon juice is $\frac{1}{4}$
of the drink), and then turn that into a percentage ($\frac{1}{4}$
into $25$%). The glass with the highest percentage being lemon
juice (or the lowest percentage being water) would be the strongest
glass of lemonade.

Will expressed the strength of the
lemonade as a percentage, as these can be easily compared.
Sharumilan, also from Wilson's School converted the fractions so
that they had a common denominator. In this way, they can be more
easily compared. In fact, this is the same as converting to
percentages; percentages are fractions with a denominator of $100$!
Here is Sharumilan's answer:

The first glass has the stronger tasting lemonade because
$\frac{2}{5}$ ($\frac{6}{15}$) of it is lemon juice while only
$\frac{1}{3}$ ($\frac{5}{15}$) of the lemonade in the second glass
is lemon juice.

What about a more visual approach? Iona,
from Whitby Maths Club compared the two solutions by drawingthese graphs.
Dominic, from Wilson's School, also suggested a graphical
approach:

If you wanted to set it out in a pie chart you could easily see
which mixture was sronger. The graph with the biggest chunk of
lemon in it would be the strongest.

Another suggestion made by several people
would be to convert the amounts so that there are equivalent
amounts of water or lemon juice, as described above. Then, you
could draw a glass and visually see which is the stronger
mixture.

Dulan, from Wilson's School, suggested a very
nice method, which can display graphically multiple different
mixtures:
He thought that a graph could be constructed with $x$ and $y$
axes. On the $x$ axis could be "amount of lemon juice", and on the
$y$ axis, "amount of water". Try constructing this for
yourself.
What does it mean if two mixtures have the same $x$ coordinate, but
different $y$ coordinates?
What if they have the same $y$ coordinates, but different $x$
coordinates?
You should be able to construct straight lines from the origin to
the various points representing different mixtures and compare
their strengths.
Along each line the strength of all of the mixtures is the same as
the proportions do not change.

We have now seen different examples of
approaches to this problem. Do the different methods always work?
Which method is most efficient?

Nathan, from Wilson's School noted that
different methods are more efficient in different situations:

The ratio or fraction strategy would be more efficient for
difficult fractions e.g. $\frac{300}{430}$ and $\frac{290}{560}$
but using a mental method would be more efficient for numbers like
$\frac{30}{50}$ and $\frac{40}{50}$, as you can see that
$\frac{40}{50}$ has a more concentrated supply of lemonade.

Several people had their own preference of
method, depending on what they felt most comfortable using.

Sharumilan explained this, and also examined
the combination of the different mixtures:

I always use fractions because it seems easier for me and the graph
helped me to get the fractions quickly so it could actually be a
mix of both.

The strength of combined mixtures always has to be the same as or
in between the two mixtures. I have said "the same as" because if
the two mixtures are identical, the combined mixture has to be the
same as well.

Try this out for yourself, with some squash,
or cordial...

Thank you very much to everyone who submitted
solutions. There were many correct solutions, and so we could not
mention them all. Well done!